Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 9 pot

Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
CHAPTER NINE
220
C A P I T A L
BUDGETING
AND RISK
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
LONG BEFORE THE development of modern theories linking risk and expected return, smart financial
managers adjusted for risk in capital budgeting. They realized intuitively that, other things being
equal, risky projects are less desirable than safe ones. Therefore, financial managers demanded a
higher rate of return from risky projects, or they based their decisions on conservative estimates of
the cash flows.
Various rules of thumb are often used to make these risk adjustments. For example, many com-
panies estimate the rate of return required by investors in their securities and then use this company
cost of capital to discount the cash flows on new projects. Our first task in this chapter is to explain
when the company cost of capital can, and cannot, be used to discount a project’s cash flows. We
shall see that it is the right hurdle rate for those projects that have the same risk as the firm’s exist-
ing business; however, if a project is more risky than the firm as a whole, the cost of capital needs to
be adjusted upward and the project’s cash flows discounted at this higher rate. Conversely, a lower

discount rate is needed for projects that are safer than the firm as a whole.
The capital asset pricing model is widely used to estimate the return that investors require.
1
It
states
We used this formula in the last chapter to figure out the return that investors expected from a sam-
ple of common stocks but we did not explain how to estimate beta. It turns out that we can gain some
insight into beta by looking at how the stock price has responded in the past to market fluctuations.
Beta is difficult to measure accurately for an individual firm: Greater accuracy can be achieved by
looking at an average of similar companies. We will also look at what features make some investments
riskier than others. If you know why Exxon Mobil has less risk than, say, Dell Computer, you will be in
a better position to judge the relative risks of different capital investment opportunities.
Some companies are financed entirely by common stock. In these cases the company cost of cap-
ital and the expected return on the stock are the same thing. However, most firms finance themselves
partly by debt and the return that they earn on their investments must be sufficient to satisfy both
the stockholders and the debtholders. We will show you how to calculate the company cost of capi-
tal when the firm has more than one type of security outstanding.
There is still another complication: Project betas can shift over time. Some projects are safer in
youth than in old age; others are riskier. In this case, what do we mean by the project beta? There
may be a separate beta for each year of the project’s life. To put it another way, can we jump from
the capital asset pricing model, which looks one period into the future, to the discounted-cash-flow
formula for valuing long-lived assets? Most of the time it is safe to do so, but you should be able to
recognize and deal with the exceptions.
We will use the capital asset pricing model, or CAPM, throughout this chapter. But don’t infer that
it is therefore the last word on risk and return. The principles and procedures covered in this chapter
work just as well with other models such as arbitrage pricing theory (APT).
Expected return ϭ r ϭ r
f
ϩ 1beta2 1r
m

Ϫ r
f
2
221
1
In a survey of financial practice, Graham and Harvey found that 74 percent of firms always, or almost always, used the capital as-
set pricing model to estimate the cost of capital. See J. Graham and C. Harvey, “The Theory and Practice of Corporate Finance: Ev-
idence from the Field,” Journal of Financial Economics 60 (May/June 2001), pp. 187–244.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
The company cost of capital is defined as the expected return on a portfolio of all
the company’s existing securities. It is used to discount the cash flows on projects
that have similar risk to that of the firm as a whole. For example, in Table 8.2 we
estimated that investors require a return of 9.2 percent from Pfizer common stock.
If Pfizer is contemplating an expansion of the firm’s existing business, it would
make sense to discount the forecasted cash flows at 9.2 percent.
2
The company cost of capital is not the correct discount rate if the new projects
are more or less risky than the firm’s existing business. Each project should in prin-
ciple be evaluated at its own opportunity cost of capital. This is a clear implication
of the value-additivity principle introduced in Chapter 7. For a firm composed of
assets A and B, the firm value is
Here PV(A) and PV(B) are valued just as if they were mini-firms in which stock-
holders could invest directly. Investors would value A by discounting its fore-
casted cash flows at a rate reflecting the risk of A. They would value B by dis-

counting at a rate reflecting the risk of B. The two discount rates will, in general, be
different. If the present value of an asset depended on the identity of the company
that bought it, present values would not add up. Remember, a good project is a
good project is a good project.
If the firm considers investing in a third project C, it should also value C as if C
were a mini-firm. That is, the firm should discount the cash flows of C at the ex-
pected rate of return that investors would demand to make a separate investment
in C. The true cost of capital depends on the use to which that capital is put.
This means that Pfizer should accept any project that more than compensates for
the project’s beta. In other words, Pfizer should accept any project lying above the
upward-sloping line that links expected return to risk in Figure 9.1. If the project
has a high risk, Pfizer needs a higher prospective return than if the project has a
low risk. Now contrast this with the company cost of capital rule, which is to ac-
cept any project regardless of its risk as long as it offers a higher return than the com-
pany’s cost of capital. In terms of Figure 9.1, the rule tells Pfizer to accept any proj-
ect above the horizontal cost of capital line, that is, any project offering a return of
more than 9.2 percent.
It is clearly silly to suggest that Pfizer should demand the same rate of return
from a very safe project as from a very risky one. If Pfizer used the company cost
of capital rule, it would reject many good low-risk projects and accept many poor
high-risk projects. It is also silly to suggest that just because another company has
a low company cost of capital, it is justified in accepting projects that Pfizer
would reject.
The notion that each company has some individual discount rate or cost of cap-
ital is widespread, but far from universal. Many firms require different returns
ϭ sum of separate asset values
Firm value ϭ PV1AB2ϭ PV1A2ϩ PV1B2
222 PART II
Risk
9.1 COMPANY AND PROJECT COSTS OF CAPITAL

2
Debt accounted for only about 0.3 percent of the total market value of Pfizer’s securities. Thus, its cost
of capital is effectively identical to the rate of return investors expect on its common stock. The compli-
cations caused by debt are discussed later in this chapter.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
from different categories of investment. For example, discount rates might be set
as follows:
CHAPTER 9
Capital Budgeting and Risk 223
Project beta
Company cost of capital
Security market line showing
required return on project
Average
beta of the firm's assets = .71
r
(required return)
3.5
9.2
FIGURE 9.1
A comparison between the
company cost of capital
rule and the required return
under the capital asset

pricing model. Pfizer’s
company cost of capital is
about 9.2 percent. This is
the correct discount rate
only if the project beta is
.71. In general, the correct
discount rate increases as
project beta increases.
Pfizer should accept
projects with rates of return
above the security market
line relating required return
to beta.
Category Discount Rate
Speculative ventures 30%
New products 20
Expansion of existing business 15 (company cost of capital)
Cost improvement, known technology 10
Perfect Pitch and the Cost of Capital
The true cost of capital depends on project risk, not on the company undertaking
the project. So why is so much time spent estimating the company cost of capital?
There are two reasons. First, many (maybe, most) projects can be treated as av-
erage risk, that is, no more or less risky than the average of the company’s other as-
sets. For these projects the company cost of capital is the right discount rate. Sec-
ond, the company cost of capital is a useful starting point for setting discount rates
for unusually risky or safe projects. It is easier to add to, or subtract from, the com-
pany cost of capital than to estimate each project’s cost of capital from scratch.
There is a good musical analogy here.
3
Most of us, lacking perfect pitch, need a

well-defined reference point, like middle C, before we can sing on key. But anyone
who can carry a tune gets relative pitches right. Businesspeople have good intuition
about relative risks, at least in industries they are used to, but not about absolute
risk or required rates of return. Therefore, they set a companywide cost of capital
as a benchmark. This is not the right hurdle rate for everything the company does,
but adjustments can be made for more or less risky ventures.
3
The analogy is borrowed from S. C. Myers and L. S. Borucki, “Discounted Cash Flow Estimates of
the Cost of Equity Capital—A Case Study,” Financial Markets, Institutions, and Investments 3 (August
1994), p. 18.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
Suppose that you are considering an across-the-board expansion by your firm. Such
an investment would have about the same degree of risk as the existing business.
Therefore you should discount the projected flows at the company cost of capital.
Companies generally start by estimating the return that investors require from
the company’s common stock. In Chapter 8 we used the capital asset pricing model
to do this. This states
An obvious way to measure the beta (␤) of a stock is to look at how its price has re-
sponded in the past to market movements. For example, look at the three left-hand
scatter diagrams in Figure 9.2. In the top-left diagram we have calculated monthly re-
turns from Dell Computer stock in the period after it went public in 1988, and we have
plotted these returns against the market returns for the same month. The second dia-
gram on the left shows a similar plot for the returns on General Motors stock, and the
third shows a plot for Exxon Mobil. In each case we have fitted a line through the

points. The slope of this line is an estimate of beta.
4
It tells us how much on average
the stock price changed for each additional 1 percent change in the market index.
The right-hand diagrams show similar plots for the same three stocks during the
subsequent period, February 1995 to July 2001. Although the slopes varied from the
first period to the second, there is little doubt that Exxon Mobil’s beta is much less
than Dell’s or that GM’s beta falls somewhere between the two. If you had used the
past beta of each stock to predict its future beta, you wouldn’t have been too far off.
Only a small portion of each stock’s total risk comes from movements in the mar-
ket. The rest is unique risk, which shows up in the scatter of points around the fitted
lines in Figure 9.2. R-squared (R
2
) measures the proportion of the total variance in the
stock’s returns that can be explained by market movements. For example, from 1995
to 2001, the R
2
for GM was .25. In other words, a quarter of GM’s risk was market risk
and three-quarters was unique risk. The variance of the returns on GM stock was 964.
5
So we could say that the variance in stock returns that was due to the market was
and the variance of unique returns was
The estimates of beta shown in Figure 9.2 are just that. They are based on the
stocks’ returns in 78 particular months. The noise in the returns can obscure the true
beta. Therefore, statisticians calculate the standard error of the estimated beta to show
the extent of possible mismeasurement. Then they set up a confidence interval of the
estimated value plus or minus two standard errors. For example, the standard error
of GM’s estimated beta in the most recent period is .20. Thus the confidence interval
for GM’s beta is 1.00 plus or minus 2 ϫ .20. If you state that the true beta for GM is
between .60 and 1.40, you have a 95 percent chance of being right. Notice that we can

be more confident of our estimate of Exxon Mobil’s beta and less confident of Dell’s.
Usually you will have more information (and thus more confidence) than this
simple calculation suggests. For example, you know that Exxon Mobil’s estimated
.75 ϫ 964 ϭ 723 25 ϫ 964 ϭ 241,
Expected stock return ϭ r
f
ϩ␤1r
m
Ϫ r
f
2
224 PART II Risk
9.2 MEASURING THE COST OF EQUITY
4
Notice that you must regress the returns on the stock on the market returns. You would get a very sim-
ilar estimate if you simply used the percentage changes in the stock price and the market index. But
sometimes analysts make the mistake of regressing the stock price level on the level of the index and ob-
tain nonsense results.
5
This is an annual figure; we annualized the monthly variance by multiplying by 12 (see footnote 17 in
Chapter 7). The standard deviation was percent.2964 ϭ 31.0
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
-10-20-30 0 10 20 30
-30

-40
-20
-10
0
10
20
30
40
50
-30 -20 -10 0 10 20 30
-40
-30
-20
-10
0
10
20
30
40
50
-30 -20 -10 0 10 20 30
-30
-20
-10
0
10
20
30
-30 -20 -10 0 10 20 30
-30

-20
-10
0
10
20
30
-30 -20 -10 0 10 20 30
-10
0
10
20
-30 -20 -10 0 10 20 30
-10
0
10
20
Dell Computer
return %
August 1988–
January 1995
β = 1.62
(.52)
R
2
= .11
Dell Computer
return %
February 1995–
July 2001
Market return, % Market return, %

β = 2.02
(.38)
R
2
= .27
August 1988–
January 1995
Market return, % Market return, %
General Motors
return %
β = .8
(.24)
R

2
= .13
February 1995–
July 2001
General Motors
return %
β = 1.00
(.20)
R
2
= .25
February 1995–
July 2001
Exxon Mobil
return %
August 1988–

January 1995
Market return, % Market return, %
β = .52
(.10)
R
2
= .28
Exxon Mobil
return %
β = .42
(.11)
R
2
= .16
FIGURE 9.2
We have used past returns to estimate the betas of three stocks for the periods August 1988 to January 1995 (left-
hand diagrams) and February 1995 to July 2001 (right-hand diagrams). Beta is the slope of the fitted line. Notice that
in both periods Dell had the highest beta and Exxon Mobil the lowest. Standard errors are in parentheses below the
betas. The standard error shows the range of possible error in the beta estimate. We also report the proportion of
total risk that is due to market movements (R
2
).
225
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003

beta was well below 1 in the previous period, while Dell’s estimated beta was well
above 1. Nevertheless, there is always a large margin for error when estimating the
beta for individual stocks.
Fortunately, the estimation errors tend to cancel out when you estimate betas of
portfolios.
6
That is why financial managers often turn to industry betas. For example,
Table 9.1 shows estimates of beta and the standard errors of these estimates for the
common stocks of four large railroad companies. Most of the standard errors are
above .2, large enough to preclude a precise estimate of any particular utility’s beta.
However, the table also shows the estimated beta for a portfolio of all four railroad
stocks. Notice that the estimated industry beta is more reliable. This shows up in
the lower standard error.
The Expected Return on Union Pacific Corporation’s Common Stock
Suppose that in mid-2001 you had been asked to estimate the company cost of cap-
ital of Union Pacific Corporation. Table 9.1 provides two clues about the true beta
of Union Pacific’s stock: the direct estimate of .40 and the average estimate for the
industry of .50. We will use the industry average of .50.
7
In mid-2001 the risk-free rate of interest r
f
was about 3.5 percent. Therefore, if
you had used 8 percent for the risk premium on the market, you would have con-
cluded that the expected return on Union Pacific’s stock was about 7.5 percent:
8
226 PART II Risk
Standard
␤
equity
Error

Burlington Northern & Santa Fe .64 .20
CSX Transportation .46 .24
Norfolk Southern .52 .26
Union Pacific Corp. .40 .21
Industry portfolio .50 .17
TABLE 9.1
Estimated betas and costs of (equity) capital for a
sample of large railroad companies and for a
portfolio of these companies. The precision of the
portfolio beta is much better than that of the
betas of the individual companies—note the lower
standard error for the portfolio.
6
If the observations are independent, the standard error of the estimated mean beta declines in propor-
tion to the square root of the number of stocks in the portfolio.
7
Comparing the beta of Union Pacific with those of the other railroads would be misleading if Union
Pacific had a materially higher or lower debt ratio. Fortunately, its debt ratio was about average for the
sample in Table 9.1.
8
This is really a discount rate for near-term cash flows, since it rests on a risk-free rate measured by the
yield on Treasury bills with maturities less than one year. Is this, you may ask, the right discount rate
for cash flows from an asset with, say, a 10- or 20-year expected life?
Well, now that you mention it, possibly not. In 2001 longer-term Treasury bonds yielded about
5.8 percent, that is, about 2.3 percent above the Treasury bill rate.
The risk-free rate could be defined as a long-term Treasury bond yield. If you do this, however,
you should subtract the risk premium of Treasury bonds over bills, which we gave as 1.8 percent in
Table 7.1. This gives a rough-and-ready estimate of the expected yield on short-term Treasury bills
over the life of the bond:
The expected average future Treasury bill rate should be used in the CAPM if a discount rate is

needed for an extended stream of cash flows. In 2001 this “long-term r
f
” was a bit higher than the
Treasury bill rate.
ϭ .058 Ϫ .019 ϭ .039, or 3.9%
Expected average T-bill rate ϭ T-bond yield Ϫ premium of bonds over bills
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
We have focused on using the capital asset pricing model to estimate the ex-
pected returns on Union Pacific’s common stock. But it would be useful to get a
check on this figure. For example, in Chapter 4 we used the constant-growth
DCF formula to estimate the expected rate of return for a sample of utility
stocks.
9
You could also use DCF models with varying future growth rates, or
perhaps arbitrage pricing theory (APT). We showed in Section 8.4 how APT can
be used to estimate expected returns.
ϭ 3.5 ϩ .518.02ϭ 7.5%
Expected stock return ϭ r
f
ϩ␤1r
m
Ϫ r
f
2

CHAPTER 9 Capital Budgeting and Risk 227
9
The United States Surface Transportation Board uses the constant-growth model to estimate the cost
of equity capital for railroad companies. We will review its findings in Chapter 19.
9.3 CAPITAL STRUCTURE AND THE COMPANY COST
OF CAPITAL
In the last section, we used the capital asset pricing model to estimate the return
that investors require from Union Pacific’s common stock. Is this figure Union Pa-
cific’s company cost of capital? Not if Union Pacific has other securities outstand-
ing. The company cost of capital also needs to reflect the returns demanded by the
owners of these securities.
We will return shortly to the problem of Union Pacific’s cost of capital, but first
we need to look at the relationship between the cost of capital and the mix of debt
and equity used to finance the company. Think again of what the company cost of
capital is and what it is used for. We define it as the opportunity cost of capital for
the firm’s existing assets; we use it to value new assets that have the same risk as
the old ones.
If you owned a portfolio of all the firm’s securities—100 percent of the debt
and 100 percent of the equity—you would own the firm’s assets lock, stock, and
barrel. You wouldn’t share the cash flows with anyone; every dollar of cash the
firm paid out would be paid to you. You can think of the company cost of capi-
tal as the expected return on this hypothetical portfolio. To calculate it, you just
take a weighted average of the expected returns on the debt and the equity:
For example, suppose that the firm’s market-value balance sheet is as follows:
Asset value 100 Debt value (D)30
Equity value (E)70
Asset value 100 Firm value (V) 100
Note that the values of debt and equity add up to the firm value (D ϩ E ϭ V) and
that the firm value equals the asset value. (These figures are market values, not book
values: The market value of the firm’s equity is often substantially different from

its book value.)
ϭ
debt
debt ϩ equity
r
debt
ϩ
equity
debt ϩ equity
r
equity
Company cost of capital ϭ r
assets
ϭ r
portfolio
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
If investors expect a return of 7.5 percent on the debt and 15 percent on the eq-
uity, then the expected return on the assets is
If the firm is contemplating investment in a project that has the same risk as the
firm’s existing business, the opportunity cost of capital for this project is the same
as the firm’s cost of capital; in other words, it is 12.75 percent.
What would happen if the firm issued an additional 10 of debt and used the cash
to repurchase 10 of its equity? The revised market-value balance sheet is
Asset value 100 Debt value (D)40

Equity value (E)60
Asset value 100 Firm value (V) 100
The change in financial structure does not affect the amount or risk of the cash
flows on the total package of debt and equity. Therefore, if investors required a re-
turn of 12.75 percent on the total package before the refinancing, they must require
a 12.75 percent return on the firm’s assets afterward.
Although the required return on the package of debt and equity is unaffected, the
change in financial structure does affect the required return on the individual se-
curities. Since the company has more debt than before, the debtholders are likely
to demand a higher interest rate. We will suppose that the expected return on the
debt rises to 7.875 percent. Now you can write down the basic equation for the re-
turn on assets
and solve for the return on equity
Increasing the amount of debt increased debtholder risk and led to a rise in the
return that debtholders required (r
debt
rose from 7.5 to 7.875 percent). The higher
leverage also made the equity riskier and increased the return that shareholders re-
quired (r
equity
rose from 15 to 16 percent). The weighted average return on debt and
equity remained at 12.75 percent:
Suppose that the company decided instead to repay all its debt and to replace it
with equity. In that case all the cash flows would go to the equity holders. The com-
pany cost of capital, r
assets
, would stay at 12.75 percent, and r
equity
would also be
12.75 percent.

ϭ 1.4 ϫ 7.8752ϩ 1.6 ϫ 162ϭ 12.75%
r
assets
ϭ 1.4 ϫ r
debt
2ϩ 1.6 ϫ r
equity
2
r
equity
ϭ 16.0%
ϭ a
40
100
ϫ 7.875 bϩ a
60
100
ϫ r
equity
bϭ 12.75%
r
assets
ϭ
D
V
r
debt
ϩ
E
V

r
equity
ϭ a
30
100
ϫ 7.5 bϩ a
70
100
ϫ 15 bϭ 12.75%
r
assets
ϭ
D
V
r
debt
ϩ
E
V
r
equity
228 PART II Risk
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
How Changing Capital Structure Affects Beta

We have looked at how changes in financial structure affect expected return. Let us
now look at the effect on beta.
The stockholders and debtholders both receive a share of the firm’s cash flows,
and both bear part of the risk. For example, if the firm’s assets turn out to be worth-
less, there will be no cash to pay stockholders or debtholders. But debtholders usu-
ally bear much less risk than stockholders. Debt betas of large blue-chip firms are
typically in the range of .1 to .3.
10
If you owned a portfolio of all the firm’s securities, you wouldn’t share the cash
flows with anyone. You wouldn’t share the risks with anyone either; you would
bear them all. Thus the firm’s asset beta is equal to the beta of a portfolio of all the
firm’s debt and its equity.
The beta of this hypothetical portfolio is just a weighted average of the debt and
equity betas:
Think back to our example. If the debt before the refinancing has a beta of .1 and
the equity has a beta of 1.1, then
What happens after the refinancing? The risk of the total package is unaffected, but
both the debt and the equity are now more risky. Suppose that the debt beta in-
creases to .2. We can work out the new equity beta:
You can see why borrowing is said to create financial leverage or gearing. Financial
leverage does not affect the risk or the expected return on the firm’s assets, but it
does push up the risk of the common stock. Shareholders demand a correspond-
ingly higher return because of this financial risk.
Figure 9.3 shows the expected return and beta of the firm’s assets. It also shows
how expected return and risk are shared between the debtholders and equity hold-
ers before the refinancing. Figure 9.4 shows what happens after the refinancing.
Both debt and equity are now more risky, and therefore investors demand a higher
return. But equity accounts for a smaller proportion of firm value than before. As
a result, the weighted average of both the expected return and beta on the two
components is unchanged.

Now you can see how to unlever betas, that is, how to go from an observed
␤
equity
to ␤
assets.
You have the equity beta, say, 1.2. You also need the debt beta, say,
.2, and the relative market values of debt (D/V) and equity (E/V). If debt accounts
for 40 percent of overall value V,
␤
assets
ϭ 1.4 ϫ .22ϩ 1.6 ϫ 1.22ϭ .8
␤
equity
ϭ 1.2
.8 ϭ 1.4 ϫ .22ϩ 1.6 ϫ␤
equity
2
␤
assets
ϭ␤
portfolio
ϭ
D
V
␤
dept
ϩ
E
V
␤

equity
␤
assets
ϭ 1.3 ϫ .12ϩ 1.7 ϫ 1.12ϭ .8
␤
assets
ϭ␤
portfolio
ϭ
D
V
␤
debt
ϩ
E
V
␤
equity
CHAPTER 9 Capital Budgeting and Risk 229
10
For example, in Table 7.1 we reported average returns on a portfolio of high-grade corporate bonds.
In the 10 years ending December 2000 the estimated beta of this bond portfolio was .17.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
This runs the previous example in reverse. Just remember the basic relationship:

Capital Structure and Discount Rates
The company cost of capital is the opportunity cost of capital for the firm’s as-
sets. That’s why we write it as r
assets.
If a firm encounters a project that has the
␤
assets
ϭ␤
portfolio
ϭ
D
V
␤
debt
ϩ
E
V
␤
equity
230 PART II Risk
Beta
0 b
debt
= .1 b
assets
= .8 b
equity
= 1.1
Expected return,
percent

r
debt
= 7.5
r
assets
= 12.75
r
equity
= 15
FIGURE 9.3
Expected returns and betas
before refinancing. The
expected return and beta
of the firm’s assets are
weighted averages of the
expected return and betas
of the debt and equity.
Beta
0 b
debt
= .2 b
assets
= .8 b
equity
= 1.2
Expected return,
percent
r
debt
= 7.875

r
assets
= 12.75
r
equity
= 16
FIGURE 9.4
Expected returns and betas
after refinancing.
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same beta as the firm’s overall assets, then r
assets
is the right discount rate for the
project cash flows.
When the firm uses debt financing, the company cost of capital is not the same
as r
equity
, the expected rate of return on the firm’s stock; r
equity
is higher because of
financial risk. However, the company cost of capital can be calculated as a
weighted average of the returns expected by investors on the various debt and eq-
uity securities issued by the firm. You can also calculate the firm’s asset beta as a
weighted average of the betas of these securities.

When the firm changes its mix of debt and equity securities, the risk and ex-
pected returns of these securities change; however, the asset beta and the company
cost of capital do not change.
Now, if you think all this is too neat and simple, you’re right. The complica-
tions are spelled out in great detail in Chapters 17 through 19. But we must note
one complication here: Interest paid on a firm’s borrowing can be deducted from
taxable income. Thus the after-tax cost of debt is r
debt
(l Ϫ T
c
), where T
c
is the
marginal corporate tax rate. When companies discount an average-risk project,
they do not use the company cost of capital as we have computed it. They use
the after-tax cost of debt to compute the after-tax weighted-average cost of cap-
ital or WACC:
More—lots more—on this in Chapter 19.
Back to Union Pacific’s Cost of Capital
In the last section we estimated the return that investors required on Union Pa-
cific’s common stock. If Union Pacific were wholly equity-financed, the company
cost of capital would be the same as the expected return on its stock. But in mid-
2001 common stock accounted for only 60 percent of the market value of the com-
pany’s securities. Debt accounted for the remaining 40 percent.
11
Union Pacific’s
company cost of capital is a weighted average of the expected returns on the dif-
ferent securities.
We estimated the expected return from Union Pacific’s common stock at 7.5 per-
cent. The yield on the company’s debt in 2001 was about 5.5 percent.

12
Thus
Union Pacific’s WACC is calculated in the same fashion, but using the after-tax cost
of debt.
ϭ a
40
100
ϫ 5.5 bϩ a
60
100
ϫ 7.5 bϭ 6.7%
Company cost of capital ϭ r
assets
ϭ
D
V
r
debt
ϩ
E
V
r
equity
WACC ϭ r
debt
11 Ϫ T
c
2
D
V

ϩ r
equity
E
V
CHAPTER 9
Capital Budgeting and Risk 231
11
Union Pacific had also issued preferred stock. Preferred stock is discussed in Chapter 14. To keep mat-
ters simple here, we have lumped the preferred stock in with Union Pacific’s debt.
12
This is a promised yield; that is, it is the yield if Union Pacific makes all the promised payments. Since
there is some risk of default, the expected return is always less than the promised yield. Union Pacific
debt has an investment-grade rating and the difference is small. But for a company that is hovering on
the brink of bankruptcy, it can be important.
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We have shown how the CAPM can help to estimate the cost of capital for domes-
tic investments by U.S. companies. But can we extend the procedure to allow for
investments in different countries? The answer is yes in principle, but naturally
there are complications.
Foreign Investments Are Not Always Riskier
Pop quiz: Which is riskier for an investor in the United States—the Standard and
Poor’s Composite Index or the stock market in Egypt? If you answer Egypt, you’re
right, but only if risk is defined as total volatility or variance. But does investment
in Egypt have a high beta? How much does it add to the risk of a diversified port-

folio held in the United States?
Table 9.2 shows estimated betas for the Egyptian market and for markets in
Poland, Thailand, and Venezuela. The standard deviations of returns in these mar-
kets were two or three times more than the U.S. market, but only Thailand had a
beta greater than 1. The reason is low correlation. For example, the standard devi-
ation of the Egyptian market was 3.1 times that of the Standard and Poor’s index,
but the correlation coefficient was only .18. The beta was 3.1 ϫ .18 ϭ .55.
Table 9.2 does not prove that investment abroad is always safer than at home.
But it should remind you always to distinguish between diversifiable and market
risk. The opportunity cost of capital should depend on market risk.
Foreign Investment in the United States
Now let’s turn the problem around. Suppose that the Swiss pharmaceutical com-
pany, Roche, is considering an investment in a new plant near Basel in Switzerland.
The financial manager forecasts the Swiss franc cash flows from the project and dis-
counts these cash flows at a discount rate measured in francs. Since the project is
risky, the company requires a higher return than the Swiss franc interest rate. How-
ever, the project is average-risk compared to Roche’s other Swiss assets. To esti-
mate the cost of capital, the Swiss manager proceeds in the same way as her coun-
terpart in a U.S. pharmaceutical company. In other words, she first measures the
risk of the investment by estimating Roche’s beta and the beta of other Swiss phar-
maceutical companies. However, she calculates these betas relative to the Swiss mar-
ket index. Suppose that both measures point to a beta of 1.1 and that the expected
232 PART II
Risk
9.4 DISCOUNT RATES FOR INTERNATIONAL
PROJECTS
Ratio of
Standard Correlation
Deviations* Coefficient Beta
†

Egypt 3.11 .18 .56
Poland 1.93 .42 .81
Thailand 2.91 .48 1.40
Venezuela 2.58 .30 .77
TABLE 9.2
Betas of four country indexes versus the U.S. market,
calculated from monthly returns, August 1996–July 2001.
Despite high volatility, three of the four betas are less
than 1. The reason is the relatively low correlation with
the U.S. market.
*Ratio of standard deviations of country index to Standard
& Poor’s Composite Index.
†
Beta is the ratio of covariance to variance. Covariance can be
written as ␴
IM
ϭ␳
IM
␴
I
␴
M
; ␤ϭ␳
IM
␴
I
␴
M
/␴
M

2
ϭ␳(␴
I
/␴
M
), where
I indicates the country index and M indicates the U.S. market.
Brealey−Meyers:
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risk premium on the Swiss market index is 6 percent.
13
Then Roche needs to dis-
count the Swiss franc cash flows from its project at 1.1 ϫ 6 ϭ 6.6 percent above the
Swiss franc interest rate.
That’s straightforward. But now suppose that Roche considers construction of a
plant in the United States. Once again the financial manager measures the risk of
this investment by its beta relative to the Swiss market index. But notice that the
value of Roche’s business in the United States is likely to be much less closely tied
to fluctuations in the Swiss market. So the beta of the U.S. project relative to the
Swiss market is likely to be less than 1.1. How much less? One useful guide is the
U.S. pharmaceutical industry beta calculated relative to the Swiss market index. It
turns out that this beta has been .36.
14
If the expected risk premium on the Swiss
market index is 6 percent, Roche should be discounting the Swiss franc cash flows

from its U.S. project at .36 ϫ 6 ϭ 2.2 percent above the Swiss franc interest rate.
Why does Roche’s manager measure the beta of its investments relative to the
Swiss index, whereas her U.S. counterpart measures the beta relative to the U.S.
index? The answer lies in Section 7.4, where we explained that risk cannot be con-
sidered in isolation; it depends on the other securities in the investor’s portfolio.
Beta measures risk relative to the investor’s portfolio. If U.S. investors already hold
the U.S. market, an additional dollar invested at home is just more of the same.
But, if Swiss investors hold the Swiss market, an investment in the United States
can reduce their risk. That explains why an investment in the United States is
likely to have lower risk for Roche’s shareholders than it has for shareholders in
Merck or Pfizer. It also explains why Roche’s shareholders are willing to accept
a lower return from such an investment than would the shareholders in the U.S.
companies.
15
When Merck measures risk relative to the U.S. market and Roche measures risk
relative to the Swiss market, their managers are implicitly assuming that the share-
holders simply hold domestic stocks. That’s not a bad approximation, particularly
in the case of the United States.
16
Although investors in the United States can re-
duce their risk by holding an internationally diversified portfolio of shares, they
generally invest only a small proportion of their money overseas. Why they are so
shy is a puzzle.
17
It looks as if they are worried about the costs of investing over-
seas, but we don’t understand what those costs include. Maybe it is more difficult
to figure out which foreign shares to buy. Or perhaps investors are worried that a
CHAPTER 9
Capital Budgeting and Risk 233
13

Figure 7.3 showed that this is the historical risk premium on the Swiss market. The fact that the real-
ized premium has been lower in Switzerland than the United States may be just a coincidence and may
not mean that Swiss investors expected the lower premium. On the other hand, if Swiss firms are gen-
erally less risky, then investors may have been content with a lower premium.
14
This is the beta of the Standard and Poor’s pharmaceutical index calculated relative to the Swiss mar-
ket for the period August 1996 to July 2001.
15
When investors hold efficient portfolios, the expected reward for risk on each stock in the portfolio is
proportional to its beta relative to the portfolio. So, if the Swiss market index is an efficient portfolio for
Swiss investors, then Swiss investors will want Roche to invest in a new plant if the expected reward
for risk is proportional to its beta relative to the Swiss market index.
16
But it can be a bad assumption elsewhere. For small countries with open financial borders—
Luxembourg, for example—a beta calculated relative to the local market has little value. Few in-
vestors in Luxembourg hold only local stocks.
17
For an explanation of the cost of capital for international investments when there are costs to interna-
tional diversification, see I. A. Cooper and E. Kaplanis, “Home Bias in Equity Portfolios and the Cost of
Capital for Multinational Firms,” Journal of Applied Corporate Finance 8 (Fall 1995), pp. 95–102.
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Risk
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Companies, 2003
foreign government will expropriate their shares, restrict dividend payments, or
catch them by a change in the tax law.
However, the world is getting smaller, and investors everywhere are increasing

their holdings of foreign securities. Large American financial institutions have sub-
stantially increased their overseas investments, and literally dozens of funds have
been set up for individuals who want to invest abroad. For example, you can now
buy funds that specialize in investment in emerging capital markets such as Viet-
nam, Peru, or Hungary. As investors increase their holdings of overseas stocks, it
becomes less appropriate to measure risk relative to the domestic market and more
important to measure the risk of any investment relative to the portfolios that they
actually hold.
Who knows? Perhaps in a few years investors will hold internationally di-
versified portfolios, and in later editions of this book we will recommend that
firms calculate betas relative to the world market. If investors throughout the
world held the world portfolio, then Roche and Merck would both demand the
same return from an investment in the United States, in Switzerland, or in
Egypt.
Do Some Countries Have a Lower Cost of Capital?
Some countries enjoy much lower rates of interest than others. For example, as we
write this the interest rate in Japan is effectively zero; in the United States it is above
3 percent. People often conclude from this that Japanese companies enjoy a lower
cost of capital.
This view is one part confusion and one part probable truth. The confusion
arises because the interest rate in Japan is measured in yen and the rate in the
United States is measured in dollars. You wouldn’t say that a 10-inch-high rabbit
was taller than a 9-foot elephant. You would be comparing their height in different
units. In the same way it makes no sense to compare an interest rate in yen with a
rate in dollars. The units are different.
But suppose that in each case you measure the interest rate in real terms. Then
you are comparing like with like, and it does make sense to ask whether the costs
of overseas investment can cause the real cost of capital to be lower in Japan. Japan-
ese citizens have for a long time been big savers, but as they moved into a new cen-
tury they were very worried about the future and were saving more than ever. That

money could not be absorbed by Japanese industry and therefore had to be in-
vested overseas. Japanese investors were not compelled to invest overseas: They
needed to be enticed to do so. So the expected real returns on Japanese investments
fell to the point that Japanese investors were willing to incur the costs of buying
foreign securities, and when a Japanese company wanted to finance a new project,
it could tap into a pool of relatively low-cost funds.
234 PART II
Risk
9.5 SETTING DISCOUNT RATES WHEN YOU
CAN’T CALCULATE BETA
Stock or industry betas provide a rough guide to the risk encountered in various
lines of business. But an asset beta for, say, the steel industry can take us only so
far. Not all investments made in the steel industry are typical. What other kinds of
evidence about business risk might a financial manager examine?
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In some cases the asset is publicly traded. If so, we can simply estimate its beta
from past price data. For example, suppose a firm wants to analyze the risks of
holding a large inventory of copper. Because copper is a standardized, widely
traded commodity, it is possible to calculate rates of return from holding copper
and to calculate a beta for copper.
What should a manager do if the asset has no such convenient price record?
What if the proposed investment is not close enough to business as usual to justify
using a company cost of capital?
These cases clearly call for judgment. For managers making that kind of judg-

ment, we offer two pieces of advice.
1. Avoid fudge factors. Don’t give in to the temptation to add fudge factors to
the discount rate to offset things that could go wrong with the proposed
investment. Adjust cash-flow forecasts first.
2. Think about the determinants of asset betas. Often the characteristics of high-
and low-beta assets can be observed when the beta itself cannot be.
Let us expand on these two points.
Avoid Fudge Factors in Discount Rates
We have defined risk, from the investor’s viewpoint, as the standard deviation of
portfolio return or the beta of a common stock or other security. But in everyday
usage risk simply equals “bad outcome.” People think of the risks of a project as a
list of things that can go wrong. For example,
• A geologist looking for oil worries about the risk of a dry hole.
• A pharmaceutical manufacturer worries about the risk that a new drug which
cures baldness may not be approved by the Food and Drug Administration.
• The owner of a hotel in a politically unstable part of the world worries about
the political risk of expropriation.
Managers often add fudge factors to discount rates to offset worries such as these.
This sort of adjustment makes us nervous. First, the bad outcomes we cited ap-
pear to reflect unique (i.e., diversifiable) risks that would not affect the expected
rate of return demanded by investors. Second, the need for a discount rate adjust-
ment usually arises because managers fail to give bad outcomes their due weight
in cash-flow forecasts. The managers then try to offset that mistake by adding a
fudge factor to the discount rate.
Example Project Z will produce just one cash flow, forecasted at $1 million at
year 1. It is regarded as average risk, suitable for discounting at a 10 percent com-
pany cost of capital:
But now you discover that the company’s engineers are behind schedule in devel-
oping the technology required for the project. They’re confident it will work, but
they admit to a small chance that it won’t. You still see the most likely outcome as

$1 million, but you also see some chance that project Z will generate zero cash flow
next year.
PV ϭ
C
1
1 ϩ r
ϭ
1,000,000
1.1
ϭ $909,100
CHAPTER 9 Capital Budgeting and Risk 235
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II. Risk 9. Capital Budgeting and
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Now the project’s prospects are clouded by your new worry about technology.
It must be worth less than the $909,100 you calculated before that worry arose. But
how much less? There is some discount rate (10 percent plus a fudge factor) that will
give the right value, but we don’t know what that adjusted discount rate is.
We suggest you reconsider your original $1 million forecast for project Z’s cash
flow. Project cash flows are supposed to be unbiased forecasts, which give due
weight to all possible outcomes, favorable and unfavorable. Managers making un-
biased forecasts are correct on average. Sometimes their forecasts will turn out
high, other times low, but their errors will average out over many projects.
If you forecast cash flow of $1 million for projects like Z, you will overestimate
the average cash flow, because every now and then you will hit a zero. Those ze-
ros should be “averaged in” to your forecasts.

For many projects, the most likely cash flow is also the unbiased forecast. If there
are three possible outcomes with the probabilities shown below, the unbiased fore-
cast is $1 million. (The unbiased forecast is the sum of the probability-weighted
cash flows.)
236 PART II
Risk
Possible Probability-Weighted Unbiased
Cash Flow Probability Cash Flow Forecast
1.2 .25 .3
1.0 .50 .5 1.0, or $1 million
.8 .25 .2
This might describe the initial prospects of project Z. But if technological uncer-
tainty introduces a 10 percent chance of a zero cash flow, the unbiased forecast
could drop to $900,000:
Possible Probability-Weighted Unbiased
Cash Flow Probability Cash Flow Forecast
1.2 .225 .27
1.0 .45 .45 .90, or $900,000
.8 .225 .18
0 .10 .0
The present value is
Now, of course, you can figure out the right fudge factor to add to the discount
rate to apply to the original $1 million forecast to get the correct answer. But you
have to think through possible cash flows to get that fudge factor; and once you
have thought through the cash flows, you don’t need the fudge factor.
Managers often work out a range of possible outcomes for major projects,
sometimes with explicit probabilities attached. We give more elaborate exam-
ples and further discussion in Chapter 10. But even when a range of outcomes
and probabilities is not explicitly written down, the manager can still consider
the good and bad outcomes as well as the most likely one. When the bad out-

comes outweigh the good, the cash-flow forecast should be reduced until bal-
ance is regained.
PV ϭ
.90
1.1
ϭ .818, or $818,000
·
·
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Step 1, then, is to do your best to make unbiased forecasts of a project’s cash
flows. Step 2 is to consider whether investors would regard the project as more or
less risky than typical for a company or division. Here our advice is to search for
characteristics of the asset that are associated with high or low betas. We wish we
had a more fundamental scientific understanding of what these characteristics are.
We see business risks surfacing in capital markets, but as yet there is no satisfac-
tory theory describing how these risks are generated. Nevertheless, some things
are known.
What Determines Asset Betas?
Cyclicality Many people intuitively associate risk with the variability of book, or
accounting, earnings. But much of this variability reflects unique or diversifiable
risk. Lone prospectors in search of gold look forward to extremely uncertain future
earnings, but whether they strike it rich is not likely to depend on the performance
of the market portfolio. Even if they do find gold, they do not bear much market
risk. Therefore, an investment in gold has a high standard deviation but a relatively

low beta.
What really counts is the strength of the relationship between the firm’s earn-
ings and the aggregate earnings on all real assets. We can measure this either by the
accounting beta or by the cash-flow beta. These are just like a real beta except that
changes in book earnings or cash flow are used in place of rates of return on secu-
rities. We would predict that firms with high accounting or cash-flow betas should
also have high stock betas—and the prediction is correct.
18
This means that cyclical firms—firms whose revenues and earnings are strongly
dependent on the state of the business cycle—tend to be high-beta firms. Thus you
should demand a higher rate of return from investments whose performance is
strongly tied to the performance of the economy.
Operating Leverage We have already seen that financial leverage (i.e., the com-
mitment to fixed-debt charges) increases the beta of an investor’s portfolio. In just
the same way, operating leverage (i.e., the commitment to fixed production charges)
must add to the beta of a capital project. Let’s see how this works.
The cash flows generated by any productive asset can be broken down into rev-
enue, fixed costs, and variable costs:
Cash flow ϭ revenue Ϫ fixed cost Ϫ variable cost
Costs are variable if they depend on the rate of output. Examples are raw materi-
als, sales commissions, and some labor and maintenance costs. Fixed costs are cash
outflows that occur regardless of whether the asset is active or idle (e.g., property
taxes or the wages of workers under contract).
We can break down the asset’s present value in the same way:
PV(asset) ϭ PV(revenue) Ϫ PV(fixed cost) Ϫ PV(variable cost)
Or equivalently
PV(revenue) ϭ PV(fixed cost) ϩ PV(variable cost) ϩ PV(asset)
CHAPTER 9
Capital Budgeting and Risk 237
18

For example, see W. H. Beaver and J. Manegold, “The Association between Market-Determined and
Accounting-Determined Measures of Systematic Risk: Some Further Evidence,” Journal of Financial and
Quantitative Analysis 10 (June 1979), pp. 231–284.
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Companies, 2003
Those who receive the fixed costs are like debtholders in the project; they simply get
a fixed payment. Those who receive the net cash flows from the asset are like hold-
ers of common stock; they get whatever is left after payment of the fixed costs.
We can now figure out how the asset’s beta is related to the betas of the values
of revenue and costs. We just use our previous formula with the betas relabeled:
In other words, the beta of the value of the revenues is simply a weighted average
of the beta of its component parts. Now the fixed-cost beta is zero by definition:
Whoever receives the fixed costs holds a safe asset. The betas of the revenues and
variable costs should be approximately the same, because they respond to the same
underlying variable, the rate of output. Therefore, we can substitute ␤
variable cost
and solve for the asset beta. Remember that ␤
fixed cost
ϭ 0.
Thus, given the cyclicality of revenues (reflected in ␤
revenue
), the asset beta is propor-
tional to the ratio of the present value of fixed costs to the present value of the project.
Now you have a rule of thumb for judging the relative risks of alternative de-
signs or technologies for producing the same project. Other things being equal, the

alternative with the higher ratio of fixed costs to project value will have the higher
project beta. Empirical tests confirm that companies with high operating leverage
actually do have high betas.
19
Searching for Clues
Recent research suggests a variety of other factors that affect an asset’s beta.
20
But
going through a long list of these possible determinants would take us too far
afield.
You cannot hope to estimate the relative risk of assets with any precision, but good
managers examine any project from a variety of angles and look for clues as to its risk-
iness. They know that high market risk is a characteristic of cyclical ventures and of
projects with high fixed costs. They think about the major uncertainties affecting the
economy and consider how projects are affected by these uncertainties.
21
ϭ␤
revenue
c1 ϩ
PV1fixed cost2
PV1asset2
d
␤
assets
ϭ␤
revenue

PV1revenue2Ϫ PV1variable cost2
PV1asset2
ϩ␤

variable cost

PV1variable cost2
PV1revenue2
ϩ␤
asset

PV1asset2
PV1revenue2
␤
revenue
ϭ␤
fixed cost

PV1fixed cost2
PV1revenue2
238 PART II Risk
19
See B. Lev, “On the Association between Operating Leverage and Risk,” Journal of Financial and Quan-
titative Analysis 9 (September 1974), pp. 627–642; and G. N. Mandelker and S. G. Rhee, “The Impact of
the Degrees of Operating and Financial Leverage on Systematic Risk of Common Stock,” Journal of Fi-
nancial and Quantitative Analysis 19 (March 1984), pp. 45–57.
20
This work is reviewed in G. Foster, Financial Statement Analysis, 2d ed., Prentice-Hall, Inc., Englewood
Cliffs, N.J., 1986, chap. 10.
21
Sharpe’s article on a “multibeta” interpretation of market risk offers a useful way of thinking about
these uncertainties and tracing their impact on a firm’s or project’s risk. See W. F. Sharpe, “The Capital
Asset Pricing Model: A ‘Multi-Beta’ Interpretation,” in H. Levy and M. Sarnat (eds.), Financial Decision
Making under Uncertainty, Academic Press, New York, 1977.

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Principles of Corporate
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II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
In practical capital budgeting, a single discount rate is usually applied to all future
cash flows. For example, the financial manager might use the capital asset pricing
model to estimate the cost of capital and then use this figure to discount each year’s
expected cash flow.
Among other things, the use of a constant discount rate assumes that project risk
does not change.
22
We know that this can’t be strictly true, for the risks to which
companies are exposed are constantly shifting. We are venturing here onto some-
what difficult ground, but there is a way to think about risk that can suggest a route
through. It involves converting the expected cash flows to certainty equivalents.
We will first explain what certainty equivalents are. Then we will use this knowl-
edge to examine when it is reasonable to assume constant risk. Finally we will
value a project whose risk does change.
Think back to the simple real estate investment that we used in Chapter 2 to intro-
duce the concept of present value. You are considering construction of an office build-
ing that you plan to sell after one year for $400,000. Since that cash flow is uncertain,
you discount at a risk-adjusted discount rate of 12 percent rather than the 7 percent
risk-free rate of interest. This gives a present value of 400,000/1.12 ϭ $357,143.
Suppose a real estate company now approaches and offers to fix the price at
which it will buy the building from you at the end of the year. This guarantee
would remove any uncertainty about the payoff on your investment. So you would
accept a lower figure than the uncertain payoff of $400,000. But how much less? If

the building has a present value of $357,143 and the interest rate is 7 percent, then
In other words, a certain cash flow of $382,143 has exactly the same present
value as an expected but uncertain cash flow of $400,000. The cash flow of $382,143
is therefore known as the certainty-equivalent cash flow. To compensate for both the
delayed payoff and the uncertainty in real estate prices, you need a return of
400,000 Ϫ 357,143 ϭ $42,857. To get rid of the risk, you would be prepared to take
a cut in the return of 400,000 Ϫ 382,143 ϭ $17,857.
Our example illustrates two ways to value a risky cash flow C
1
:
Method 1: Discount the risky cash flow at a risk-adjusted discount rate r that is
greater than r
f
.
23
The risk-adjusted discount rate adjusts for both time and risk.
This is illustrated by the clockwise route in Figure 9.5.
Method 2: Find the certainty-equivalent cash flow and discount at the risk-free
interest rate r
f
. When you use this method, you need to ask, What is the
smallest certain payoff for which I would exchange the risky cash flow C
1
?
Certain cash flow ϭ $382,143
PV ϭ
Certain cash flow
1.07
ϭ $357,143
CHAPTER 9 Capital Budgeting and Risk 239

9.6 ANOTHER LOOK AT RISK AND DISCOUNTED
CASH FLOW
22
See E. F. Fama, “Risk-Adjusted Discount Rates and Capital Budgeting under Uncertainty,” Journal of
Financial Economics 5 (August 1977), pp. 3–24; or S. C. Myers and S. M. Turnbull, “Capital Budgeting
and the Capital Asset Pricing Model: Good News and Bad News,” Journal of Finance 32 (May 1977),
pp. 321–332.
23
The quantity r can be less than r
f
for assets with negative betas. But the betas of the assets that corpo-
rations hold are almost always positive.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
This is called the certainty equivalent of C
1
denoted by CEQ
1.
24
Since CEQ
1
is
the value equivalent of a safe cash flow, it is discounted at the risk-free rate.
The certainty-equivalent method makes separate adjustments for risk and time.
This is illustrated by the counterclockwise route in Figure 9.5.

We now have two identical expressions for PV:
For cash flows two, three, or t years away,
When to Use a Single Risk-Adjusted Discount Rate for Long-Lived Assets
We are now in a position to examine what is implied when a constant risk-adjusted
discount rate, r, is used to calculate a present value.
Consider two simple projects. Project A is expected to produce a cash flow of
$100 million for each of three years. The risk-free interest rate is 6 percent, the mar-
ket risk premium is 8 percent, and project A’s beta is .75. You therefore calculate A’s
opportunity cost of capital as follows:
Discounting at 12 percent gives the following present value for each cash flow:
ϭ 6 ϩ .75182ϭ 12%
r ϭ r
f
ϩ␤1r
m
Ϫ r
f
2
PV ϭ
C
t
11 ϩ r2
t
ϭ
CEQ
t
11 ϩ r
f
2
t

PV ϭ
C
1
1 ϩ r
ϭ
CEQ
1
1 ϩ r
f
240 PART II Risk
Risk-Adjusted Discount Rate Method
Certainty-Equivalent Method
Discount for time and risk
Present
value
Future
cash
flow
C
1
Discount for time
value of money
Haircut
for risk
FIGURE 9.5
Two ways to
calculate present
value. “Haircut for
risk” refers to the
reduction of the cash

flow from its fore-
casted value to its
certainty equivalent.
24
CEQ
1
can be calculated directly from the capital asset pricing model. The certainty-equivalent form
of the CAPM states that the certainty-equivalent value of the cash flow, C
l
, is PV ϭ C
l
Ϫ␭cov(
˜
C
l
, ˜r
m
).
Cov (
˜
C
1
, ˜r
m
) is the covariance between the uncertain cash flow,
˜
C
1
, and the return on the market, r
m

.
Lambda, ␭, is a measure of the market price of risk. It is defined as (r
m
Ϫ r
f
)/␴
m
2
. For example, if r
m
Ϫ
r
f
ϭ .08 and the standard deviation of market returns is ␴
m
ϭ .20, then lambda ϭ .08/.20
2
ϭ 2. We show
on the Brealey-Myers website (www.mhhe.com/bm7e) how the CAPM formula can be twisted around
into this certainty-equivalent form.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
CHAPTER 9 Capital Budgeting and Risk 241
Project A
Year Cash Flow PV at 12%

1 100 89.3
2 100 79.7
3 100 71.2
Total PV 240.2
Now compare these figures with the cash flows of project B. Notice that B’s cash
flows are lower than A’s; but B’s flows are safe, and therefore they are discounted
at the risk-free interest rate. The present value of each year’s cash flow is identical
for the two projects.
Project B
Year Cash Flow PV at 6%
1 94.6 89.3
2 89.6 79.7
3 84.8 71.2
Total PV 240.2
In year 1 project A has a risky cash flow of 100. This has the same PV as the safe
cash flow of 94.6 from project B. Therefore 94.6 is the certainty equivalent of 100.
Since the two cash flows have the same PV, investors must be willing to give up
100 Ϫ 94.6 ϭ 5.4 in expected year-1 income in order to get rid of the uncertainty.
In year 2 project A has a risky cash flow of 100, and B has a safe cash flow of 89.6.
Again both flows have the same PV. Thus, to eliminate the uncertainty in year 2,
investors are prepared to give up 100 Ϫ 89.6 ϭ 10.4 of future income. To eliminate
uncertainty in year 3, they are willing to give up 100 Ϫ 84.8 ϭ 15.2 of future income.
To value project A, you discounted each cash flow at the same risk-adjusted dis-
count rate of 12 percent. Now you can see what is implied when you did that. By
using a constant rate, you effectively made a larger deduction for risk from the later
cash flows:
Forecasted Certainty-
Cash Flow for Equivalent Deduction
Year Project A Cash Flow for Risk
1 100 94.6 5.4

2 100 89.6 10.4
3 100 84.8 15.2
The second cash flow is riskier than the first because it is exposed to two years of mar-
ket risk. The third cash flow is riskier still because it is exposed to three years of mar-
ket risk. This increased risk is reflected in the steadily declining certainty equivalents:
Forecasted Certainty-
Cash Flow for Equivalent Ratio of CEQ
t
Year Project A (C
t
) Cash Flow (CEQ
t
) to C
t
1 100 94.6 .946
2 100 89.6 .896 ϭ .946
2
3 100 84.8 .848 ϭ .946
3
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
Our example illustrates that if we are to use the same discount rate for every future
cash flow, then the certainty equivalents must decline steadily as a fraction of the cash
flow. There’s no law of nature stating that certainty equivalents have to decrease in
this smooth and regular way. It may be a fair assumption for most projects most of the

time, but we’ll sketch in a moment a real example in which that is not the case.
A Common Mistake
You sometimes hear people say that because distant cash flows are riskier, they
should be discounted at a higher rate than earlier cash flows. That is quite wrong: We
have just seen that using the same risk-adjusted discount rate for each year’s cash
flow implies a larger deduction for risk from the later cash flows. The reason is that
the discount rate compensates for the risk borne per period. The more distant the cash
flows, the greater the number of periods and the larger the total risk adjustment.
When You Cannot Use a Single Risk-Adjusted Discount Rate
for Long-Lived Assets
Sometimes you will encounter problems where risk does change as time passes, and
the use of a single risk-adjusted discount rate will then get you into trouble. For ex-
ample, later in the book we will look at how options are valued. Because an option’s
risk is continually changing, the certainty-equivalent method needs to be used.
Here is a disguised, simplified, and somewhat exaggerated version of an actual
project proposal that one of the authors was asked to analyze. The scientists at Veg-
etron have come up with an electric mop, and the firm is ready to go ahead with pi-
lot production and test marketing. The preliminary phase will take one year and cost
$125,000. Management feels that there is only a 50 percent chance that pilot produc-
tion and market tests will be successful. If they are, then Vegetron will build a $1 mil-
lion plant that would generate an expected annual cash flow in perpetuity of $250,000
a year after taxes. If they are not successful, the project will have to be dropped.
The expected cash flows (in thousands of dollars) are
Management has little experience with consumer products and considers this a
project of extremely high risk.
25
Therefore management discounts the cash flows
at 25 percent, rather than at Vegetron’s normal 10 percent standard:
This seems to show that the project is not worthwhile.
Management’s analysis is open to criticism if the first year’s experiment resolves a

high proportion of the risk. If the test phase is a failure, then there’s no risk at all—the
project is certain to be worthless. If it is a success, there could well be only normal risk
from then on. That means there is a 50 percent chance that in one year Vegetron will
NPV ϭϪ125 Ϫ
500
1.25
ϩ
a
∞
tϭ2
125
11.252
t
ϭϪ125, or Ϫ$125,000
ϭ .512502ϩ .5102ϭ 125
C
t
for t ϭ 2, 3,
…
ϭ 50% chance of 250 and 50% chance of 0
ϭ .51Ϫ1,0002ϩ .5102ϭϪ500
C
l
ϭ 50% chance of Ϫ1,000 and 50% chance of 0
C
0
ϭϪ125
242 PART II Risk
25
We will assume that they mean high market risk and that the difference between 25 and 10 percent is

not a fudge factor introduced to offset optimistic cash-flow forecasts.
Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
CHAPTER 9 Capital Budgeting and Risk 243
have the opportunity to invest in a project of normal risk, for which the normal dis-
count rate of 10 percent would be appropriate. Thus the firm has a 50 percent chance
to invest $1 million in a project with a net present value of $1.5 million:
Success
—→ (50% chance)
Pilot production
and market tests
Failure —→ NPV ϭ 0 (50% chance)
Thus we could view the project as offering an expected payoff of .5(1,500)ϩ.5(0) ϭ
750, or $750,000, at t ϭ 1 on a $125,000 investment at t ϭ 0. Of course, the certainty
equivalent of the payoff is less than $750,000, but the difference would have to be very
large to justify rejecting the project. For example, if the certainty equivalent is half the
forecasted cash flow and the risk-free rate is 7 percent, the project is worth $225,500:
This is not bad for a $125,000 investment—and quite a change from the negative-
NPV that management got by discounting all future cash flows at 25 percent.
ϭϪ125 ϩ
.517502
1.07
ϭ 225.5, or $225,500
NPV ϭ C
0

ϩ
CEQ
1
1 ϩ r
NPV ϭϪ1000 ϩ
250
.10
ϭϩ1,500
SUMMARY
In Chapter 8 we set out some basic principles for valuing risky assets. In this chap-
ter we have shown you how to apply these principles to practical situations.
The problem is easiest when you believe that the project has the same market
risk as the company’s existing assets. In this case, the required return equals the re-
quired return on a portfolio of all the company’s existing securities. This is called
the company cost of capital.
Common sense tells us that the required return on any asset depends on its risk.
In this chapter we have defined risk as beta and used the capital asset pricing
model to calculate expected returns.
The most common way to estimate the beta of a stock is to figure out how the
stock price has responded to market changes in the past. Of course, this will give
you only an estimate of the stock’s true beta. You may get a more reliable figure if
you calculate an industry beta for a group of similar companies.
Suppose that you now have an estimate of the stock’s beta. Can you plug that
into the capital asset pricing model to find the company’s cost of capital? No, the
stock beta may reflect both business and financial risk. Whenever a company bor-
rows money, it increases the beta (and the expected return) of its stock. Remember,
the company cost of capital is the expected return on a portfolio of all the firm’s se-
curities, not just the common stock. You can calculate it by estimating the expected
return on each of the securities and then taking a weighted average of these sepa-
rate returns. Or you can calculate the beta of the portfolio of securities and then

plug this asset beta into the capital asset pricing model.
The company cost of capital is the correct discount rate for projects that have the
same risk as the company’s existing business. Many firms, however, use the com-
pany cost of capital to discount the forecasted cash flows on all new projects. This
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Brealey−Meyers:
Principles of Corporate
Finance, Seventh Edition
II. Risk 9. Capital Budgeting and
Risk
© The McGraw−Hill
Companies, 2003
There is a good review article by Rubinstein on the application of the capital asset pricing model to
capital investment decisions:
M. E. Rubinstein: “A Mean-Variance Synthesis of Corporate Financial Theory,” Journal of Fi-
nance, 28:167–182 (March 1973).
There have been a number of studies of the relationship between accounting data and beta. Many of
these are reviewed in:
G. Foster: Financial Statement Analysis, 2nd ed., Prentice-Hall, Inc., Englewood Cliffs,
N.J., 1986.
244 PART II Risk
is a dangerous procedure. In principle, each project should be evaluated at its own
opportunity cost of capital; the true cost of capital depends on the use to which the
capital is put. If we wish to estimate the cost of capital for a particular project, it is
project risk that counts. Of course the company cost of capital is fine as a discount
rate for average-risk projects. It is also a useful starting point for estimating dis-
count rates for safer or riskier projects.
These basic principles apply internationally, but of course there are complications.
The risk of a stock or real asset may depend on who’s investing. For example, a Swiss
investor would calculate a lower beta for Merck than an investor in the United States.

Conversely, the U.S. investor would calculate a lower beta for a Swiss pharmaceuti-
cal company than a Swiss investor. Both investors see lower risk abroad because of
the less-than-perfect correlation between the two countries’ markets.
If all investors held the world market portfolio, none of this would matter. But there
is a strong home-country bias. Perhaps some investors stay at home because they re-
gard foreign investment as risky. We suspect they confuse total risk with market risk.
For example, we showed examples of countries with extremely volatile stock mar-
kets. Most of these markets were nevertheless low-beta investments for an investor
holding the U.S. market. Again, the reason was low correlation between markets.
Then we turned to the problem of assessing project risk. We provided several
clues for managers seeking project betas. First, avoid adding fudge factors to dis-
count rates to offset worries about bad project outcomes. Adjust cash-flow forecasts
to give due weight to bad outcomes as well as good; then ask whether the chance of
bad outcomes adds to the project’s market risk. Second, you can often identify the
characteristics of a high- or low-beta project even when the project beta cannot be cal-
culated directly. For example, you can try to figure out how much the cash flows are
affected by the overall performance of the economy: Cyclical investments are gener-
ally high-beta investments. You can also look at the project’s operating leverage:
Fixed production charges work like fixed debt charges; that is, they increase beta.
There is one more fence to jump. Most projects produce cash flows for several
years. Firms generally use the same risk-adjusted rate to discount each of these
cash flows. When they do this, they are implicitly assuming that cumulative risk
increases at a constant rate as you look further into the future. That assumption is
usually reasonable. It is precisely true when the project’s future beta will be con-
stant, that is, when risk per period is constant.
But exceptions sometimes prove the rule. Be on the alert for projects where risk
clearly does not increase steadily. In these cases, you should break the project into
segments within which the same discount rate can be reasonably used. Or you
should use the certainty-equivalent version of the DCF model, which allows sepa-
rate risk adjustments to each period’s cash flow.

FURTHER
READING
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